Theorem inimass 6116

Description: The image of an intersection. (Contributed by Thierry Arnoux, 16-Dec-2017.)

Assertion

Ref	Expression
inimass	⊢ ((𝐴 ∩ 𝐵) “ 𝐶) ⊆ ((𝐴 “ 𝐶) ∩ (𝐵 “ 𝐶))

Proof of Theorem inimass

Step	Hyp	Ref	Expression
1		rnin 6107	. 2 ⊢ ran ((𝐴 ↾ 𝐶) ∩ (𝐵 ↾ 𝐶)) ⊆ (ran (𝐴 ↾ 𝐶) ∩ ran (𝐵 ↾ 𝐶))
2		df-ima 5644	. . 3 ⊢ ((𝐴 ∩ 𝐵) “ 𝐶) = ran ((𝐴 ∩ 𝐵) ↾ 𝐶)
3		resindir 5956	. . . 4 ⊢ ((𝐴 ∩ 𝐵) ↾ 𝐶) = ((𝐴 ↾ 𝐶) ∩ (𝐵 ↾ 𝐶))
4	3	rneqi 5890	. . 3 ⊢ ran ((𝐴 ∩ 𝐵) ↾ 𝐶) = ran ((𝐴 ↾ 𝐶) ∩ (𝐵 ↾ 𝐶))
5	2, 4	eqtri 2752	. 2 ⊢ ((𝐴 ∩ 𝐵) “ 𝐶) = ran ((𝐴 ↾ 𝐶) ∩ (𝐵 ↾ 𝐶))
6		df-ima 5644	. . 3 ⊢ (𝐴 “ 𝐶) = ran (𝐴 ↾ 𝐶)
7		df-ima 5644	. . 3 ⊢ (𝐵 “ 𝐶) = ran (𝐵 ↾ 𝐶)
8	6, 7	ineq12i 4177	. 2 ⊢ ((𝐴 “ 𝐶) ∩ (𝐵 “ 𝐶)) = (ran (𝐴 ↾ 𝐶) ∩ ran (𝐵 ↾ 𝐶))
9	1, 5, 8	3sstr4i 3995	1 ⊢ ((𝐴 ∩ 𝐵) “ 𝐶) ⊆ ((𝐴 “ 𝐶) ∩ (𝐵 “ 𝐶))

Syntax hints:   ∩ cin 3910   ⊆ wss 3911  ran crn 5632   ↾ cres 5633   “ cima 5634

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-br 5103  df-opab 5165  df-xp 5637  df-rel 5638  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644

This theorem is referenced by:  restutopopn  24102